Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(435\)\(\medspace = 3 \cdot 5 \cdot 29 \) |
| Artin field: | Galois closure of \(\Q(\sqrt{870 -150 \sqrt{29}})\) |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{435}(104,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 112x^{2} + 328x + 139 \)
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The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 13\cdot 53 + 14\cdot 53^{2} + 3\cdot 53^{3} + 12\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 14\cdot 53 + 13\cdot 53^{2} + 46\cdot 53^{3} + 4\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 + 52\cdot 53 + 22\cdot 53^{2} + 40\cdot 53^{3} + 39\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 48 + 25\cdot 53 + 2\cdot 53^{2} + 16\cdot 53^{3} + 49\cdot 53^{4} +O(53^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-1$ | |
| $1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ | |
| $1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ |